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Reconstruction from noisy projections

Tomographic reconstruction using algebraic techniques is complicated by the presence of noise in the collected data much in the same way as the filtered back projection method. Within this project, we have investigated two alternative filtering methods for suppressing the effects of this noise.

First, we investigated the possibility of including a Wiener filter as the constraining operator in the minimization problem (17), as derived by Jain [5] and Castleman [1]. As shown by these authors, the resulting equation is

Here, is the noise-to-signal ratio, and is the inverse of the autocorrelation matrix for the image. Since the matrix R is the same size as , it is not feasible to compute the inverse of this matrix exactly. We therefore attempted to approximate by a sparse matrix retaining only the largest coefficients. This approximation resulted in a matrix Q similar to that used for the Laplacian smoothing constraint described by Castleman [1]. We were unable, however, to obtain useful results with this method, presumably due to errors introduced in approximating the matrix.

We turned therefore to a method similar to that used for the filtered back-projection, where the projection data were 1-D filtered using a Wiener smoothing filter based upon the same image model as in the filtered back-projection method. For this method, however, the required filter is simply a lowpass filter, given by

since the ramp portion of the Wiener filter used for filtered back-projection is a part of that particular reconstruction method. The frequency response of the lowpass filter used for the algebraic technique is shown in Fig. 10, along with the resulting reconstructed image.

  
Figure 10: The frequency response of the Wiener filter used for the algebraic reconstruction method (left), and the resulting reconstructed image (right).


next up previous
Next: Summary Up: Algebraic Reconstruction Methods Previous: Solution methods for the

Anders Johan Nygren
Thu May 8 12:28:25 CDT 1997